I'm trying to implement the circle spline scheme described here. I feel like I have a solid working understanding of the procedure, but I'm generally pretty rusty when it comes to geometry. A big sticking point is in section 3.2 (at the top of page 205), where a function for a tangent vector, $t(u)$, is described. It's "found by a swivel operation" in a plane and drawn at a point. I've tried to look up what "swivel operation" means to no avail; my current assumption is that it means rotating a vector around an axis by an angle.
However, that assumption might be wrong, because my current implementation just sweeps a line between each pair of points, instead of the smooth circle-like curve it should produce. Even if the assumption is correct, I don't see where the paper explains which vector should be rotated to begin with (I've tried $\mathbf{b}$ and $t_i$, but neither seemed to make a difference).
I think I'm approaching $t(u)$ wrong, so I'll try to start over here. What is the procedure they are describing to find $t(u)$?